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Most survey sampling estimators derive their large sample properties by establishing an asymptotic equivalence to the Horvitz-Thompson (HT) estimator If the proposed estimator is asymptotically equivalent to the HT estimator, then it inherits the HT estimator asymptotic properties such as design consistency with a limiting normal distribution. Although this approach is valid, it does not provide insights on the proposed estimator’s efficiency in small samples. As a result, most papers include simulation studies to examine these properties empirically.
We take a different approach and show that methods from classical asymptotic theory for estimators as functions of random variables can be used to derive the asymptotic property of survey sampling estimators. The focus of this approach is the discrete random vector of the sample membership indicators as the only stochastic component of the estimator. The use of discrete multivariate statistics and matrix operations reduces the derivation of the expressions of the estimator and its asymptotic properties to an algebraic problem while providing new insights into its properties. We illustrate these methods by deriving the variance, variance estimator, and determining the sufficient conditions for the HT estimator and its variance estimator to be design consistent.
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